1. 10.2 The Ellipse
Psalm 63:5 "My soul will be satisfied as with fat and rich
food, and my mouth will praise you with joyful lips,"

2. Another way to define
the ellipse is this:
An Ellipse is the set of
points in the plane the
sum of whose distances
from two fixed points
(the Foci), is a constant.
use K on keyboard to Pause/Play

3. Another way to define
the ellipse is this:
An Ellipse is the set of
points in the plane the
sum of whose distances
from two fixed points
(the Foci), is a constant.
How narrow or wide the ellipse is will be
called its eccentricity. More on this later.
use K on keyboard to Pause/Play

4. Another way to define
the ellipse is this:
An Ellipse is the set of
points in the plane the
sum of whose distances
from two fixed points
(the Foci), is a constant.
How narrow or wide the ellipse is will be
called its eccentricity. More on this later.
use K on keyboard to Pause/Play

5. What follows is the derivation of the equation
for an ellipse centered at the origin.

6. What follows is the derivation of the equation
for an ellipse centered at the origin.
You don’t need to write any of this down, but I
want you to pay careful attention and
understand the work.

7. Consider this ellipse centered at the Origin with foci on
the x-axis:
and let the sum of F1P and F2 P be the constant 2a
which means ± a are the x-intercepts and a > c

8. Consider this ellipse centered at the Origin with foci on
the x-axis:
and let the sum of F1P and F2 P be the constant 2a
which means ± a are the x-intercepts and a > c
then
2 2 2 2
( x + c) +y + ( x − c) + y = 2a

9. from the previous slide
2 2 2 2
( x + c) +y + ( x − c) + y = 2a

10. from the previous slide
2 2 2 2
( x + c) +y + ( x − c) + y = 2a
2 2 2 2
( x − c) + y = 2a − ( x + c) +y

11. from the previous slide
2 2 2 2
( x + c) +y + ( x − c) + y = 2a
2 2 2 2
( x − c) + y = 2a − ( x + c) +y
square both sides and expand

12. from the previous slide
2 2 2 2
( x + c) +y + ( x − c) + y = 2a
2 2 2 2
( x − c) + y = 2a − ( x + c) +y
square both sides and expand
2 2 2 2 2 2 2 2 2
x − 2cx + c + y = 4a − 4a ( x + c) + y + x + 2cx + c + y

13. from the previous slide
2 2 2 2
( x + c) +y + ( x − c) + y = 2a
2 2 2 2
( x − c) + y = 2a − ( x + c) +y
square both sides and expand
2 2 2 2 2 2 2 2 2
x − 2cx + c + y = 4a − 4a ( x + c) + y + x + 2cx + c + y

14. from the previous slide
2 2 2 2
( x + c) +y + ( x − c) + y = 2a
2 2 2 2
( x − c) + y = 2a − ( x + c) +y
square both sides and expand
2 2 2 2 2 2 2 2 2
x − 2cx + c + y = 4a − 4a ( x + c) + y + x + 2cx + c + y
2 2 2
4a ( x + c) + y = 4a + 4cx

15. from the previous slide
2 2 2 2
( x + c) +y + ( x − c) + y = 2a
2 2 2 2
( x − c) + y = 2a − ( x + c) +y
square both sides and expand
2 2 2 2 2 2 2 2 2
x − 2cx + c + y = 4a − 4a ( x + c) + y + x + 2cx + c + y
2 2 2
4a ( x + c) + y = 4a + 4cx
2 2 2
a ( x + c) + y = a + cx

16. from the previous slide
2 2 2
a ( x + c) + y = a + cx

17. from the previous slide
2 2 2
a ( x + c) + y = a + cx
square both sides

18. from the previous slide
2 2 2
a ( x + c) + y = a + cx
square both sides
a 2
(( x + c ) + y ) = a
2 2 4 2 2
+ 2a cx + c x 2

19. from the previous slide
2 2 2
a ( x + c) + y = a + cx
square both sides
a 2
(( x + c ) + y ) = a
2 2 4 2
+ 2a cx + c x 2 2
2 2 2 2 2 2 2 4 2 2 2
a x + 2a cx + a c + a y = a + 2a cx + c x

20. from the previous slide
2 2 2
a ( x + c) + y = a + cx
square both sides
a 2
(( x + c ) + y ) = a
2 2 4 2
+ 2a cx + c x 2 2
2 2 2 2 2 2 2 4 2 2 2
a x + 2a cx + a c + a y = a + 2a cx + c x

21. from the previous slide
2 2 2
a ( x + c) + y = a + cx
square both sides
a 2
(( x + c ) + y ) = a
2 2 4 2
+ 2a cx + c x 2 2
2 2 2 2 2 2 2 4 2 2 2
a x + 2a cx + a c + a y = a + 2a cx + c x
2 2 2 2 2 2 4 2 2
a x +a c +a y =a +c x

22. from the previous slide
2 2 2
a ( x + c) + y = a + cx
square both sides
a 2
(( x + c ) + y ) = a
2 2 4 2
+ 2a cx + c x 2 2
2 2 2 2 2 2 2 4 2 2 2
a x + 2a cx + a c + a y = a + 2a cx + c x
2 2 2 2 2 2 4 2 2
a x +a c +a y =a +c x
subtract each of these terms from both sides and group

23. from the previous slide
2 2 2
a ( x + c) + y = a + cx
square both sides
a 2
(( x + c ) + y ) = a
2 2 4 2
+ 2a cx + c x 2 2
2 2 2 2 2 2 2 4 2 2 2
a x + 2a cx + a c + a y = a + 2a cx + c x
2 2 2 2 2 2 4 2 2
a x +a c +a y =a +c x
subtract each of these terms from both sides and group
(a x
2 2
− c x )+ a y = a − a c
2 2 2 2 4 2 2

24. from the previous slide
2 2 2
a ( x + c) + y = a + cx
square both sides
a 2
(( x + c ) + y ) = a
2 2 4 2
+ 2a cx + c x 2 2
2 2 2 2 2 2 2 4 2 2 2
a x + 2a cx + a c + a y = a + 2a cx + c x
2 2 2 2 2 2 4 2 2
a x +a c +a y =a +c x
subtract each of these terms from both sides and group
(a x 2 2
− c x )+ a y = a − a c
2 2 2 2 4 2 2
x (a
2 2
− c ) + a y = a (a − c
2 2 2 2 2 2
)

25. from the previous slide
x (a − c ) + a y = a (a − c
2 2 2 2 2 2 2 2
)

26. from the previous slide
x (a − c ) + a y = a (a − c
2 2 2 2 2 2 2 2
)
recall that a > c , so a − c > 0 and
2 2
2
(
we can divide by a a − c2 2
)

27. from the previous slide
x (a − c ) + a y = a (a − c
2 2 2 2 2 2 2 2
)
recall that a > c , so a − c > 0 and
2 2
2
(
we can divide by a a − c2 2
)
2 2
x y
2
+ 2 2 =1
a a −c

28. from the previous slide
x (a − c ) + a y = a (a − c
2 2 2 2 2 2 2 2
)
recall that a > c , so a − c > 0 and
2 2
we can divide by a a − c
2 2
(2
)
2 2
x y
2
+ 2 2 =1
a a −c
then we let b = a − c
2 2 2
( and b < a )
to get the familiar:

29. from the previous slide
x (a − c ) + a y = a (a − c
2 2 2 2 2 2 2 2
)
recall that a > c , so a − c > 0 and 2 2
we can divide by a a − c
2 2 2
( )
2 2
x y
2
+ 2 2 =1
a a −c
then we let b = a − c 2 2 2
( and b < a )
to get the familiar:
2 2
x y
2
+ 2 =1
a b

30. 2 2
x y
2
+ 2 =1
a b
major axis has length 2a
minor axis has length 2b

31. 2 2
x y
2
+ 2 =1
b a
major axis has length 2a
minor axis has length 2b

32. To sketch the graph of an ellipse, find the lengths of
the major and minor axes ... and sketch away! Please
include the foci.

33. Example: Sketch the graph of 4x 2 + 25y 2 = 100

34. Example: Sketch the graph of 4x 2 + 25y 2 = 100
divide by the constant to put it into standard form

35. Example: Sketch the graph of 4x 2 + 25y 2 = 100
divide by the constant to put it into standard form
2 2
x y
+ =1
25 4

36. Example: Sketch the graph of 4x 2 + 25y 2 = 100
divide by the constant to put it into standard form
2 2
x y
+ =1
25 4
2
a = 25
a = ±5

37. Example: Sketch the graph of 4x 2 + 25y 2 = 100
divide by the constant to put it into standard form
2 2
x y
+ =1
25 4
2 2
a = 25 b =4
a = ±5 b = ±2

38. Example: Sketch the graph of 4x 2 + 25y 2 = 100
divide by the constant to put it into standard form
2 2
x y
+ =1
25 4
2 2 2 2 2
a = 25 b =4 c = a −b
2
a = ±5 b = ±2 c = 25 − 4
c = ± 21

39. Example: Sketch the graph of 4x 2 + 25y 2 = 100
divide by the constant to put it into standard form
2 2
x y
+ =1
25 4
2 2 2 2 2
a = 25 b =4 c = a −b
2
a = ±5 b = ±2 c = 25 − 4
c = ± 21

40. HW #3
“As a rule of thumb, involve everyone in everything.”
Tom Peters